Analytical insights into the transient climate response

Abstract

The temperature in the transient climate response is lower than the equilibrium temperature for the same amount of forcing. The degree of disequilibrium is not constant in time and depends on various climate parameters. We derive intuition for this by solving the heat equation with a surface temperature feedback for linearly increasing forcing. The surface temperature initially evolves at a slower rate than the corresponding steady state (SS) temperature and it accelerates until quasi-steady state (QSS), when the SS and QSS temperatures evolve in parallel with a constant offset. The offset depends on the rate of forcing and total heat capacity of the system divided by the square of the climate feedback. The timescale over which the climate system approaches QSS depends also on the effective ocean mixing and is order of thousands of years. Over societally relevant timescales (around 100 years), the top-of-atmosphere energy imbalance increases, and the actual temperature moves farther from the steady-state temperature expected for the same forcing.

Figures (5)

• Figure 1: Analytical (slowest decaying mode only) and numerical (all decaying modes) solutions to the heat equation with parameters $F_{2\times}=3.5$ Wm$^{-2}$, $\lambda=1$ Wm$^{-2}$K$^{-1}$, $K=350$ Wm$^{-1}$K$^{-1}$, $L=2600$ m and a forcing increase of 1 permille/year. Evolution of surface temperature anomaly (a), corresponding temperature profiles in depth (b) - individual gray lines are 100 years apart, evolution of surface temperature anomaly relative to the steady state (SS) temperature (c), evolution of the forcing (d).
• Figure 2: Temporal evolution of the apparent heat capacity (a) and top of domain imbalance (b). Phase space of accumulated energy and surface temperature (c). Parameters are the same as in Figure \ref{['fig:ts']}.
• Figure 3: Temporal evolution of surface temperature (a), top of atmosphere imbalance (b), effective heat capacity (c) and accumulated energy and surface temperature phase space (d). Blue lines correspond to the GFDL-CM4 1pctCO2 run (anomalies relative to piControl). Orange lines are the analytical approximation (equations \ref{['eq:qss_approach']}-\ref{['eq:qss_approachn']}) with $F_{2x}=3.13$ Wm$^{-2}$, $\lambda=0.83$ Wm$^{-2}$K$^{-1}$, $K=318$ Wm$^{-1}$K$^{-1}$, $L=2600$ m.
• Figure 4: Numerical solutions of the heat equation and the 2-layer model. Parameters for the heat equation are the same as in Figure \ref{['fig:ts']}; equivalent parameters parameters for the 2-box model are chosen for $k=0.1$ and $L=2600$ m as $C_1=260$ m, $C_2=2340$ m, the equivalent $\gamma$ is estimated to be $0.38$ Wm$^{-2}$K$^{-1}$.
• Figure 5: The apparent heat capacity as equivalent depth for different functional forms of the forcing diagnosed from the numerical solution of the heat equation (left) and diagnosed from the two-layer model (right). Model parameters are the same as in Figure \ref{['fig:2box']}. $dT_2/dT_1$ is calculated from equation \ref{['eq:dt2dt1']} with $\epsilon=1$.